Planar convex codes are decidable

TL;DR

This paper proves that determining whether a convex code can be realized in the plane with polygons is decidable, providing bounds on complexity and an explicit algorithm for the decision problem.

Contribution

It establishes the decidability of planar convex code realizations and introduces bounds on the number of vertices needed for such realizations.

Findings

Existence of polygonal realizations for convex codes in the plane

Factorial bounds on the number of vertices needed

Algorithm for deciding realizability in the plane

Abstract

We show that every convex code realizable by compact sets in the plane admits a realization consisting of polygons, and analogously every open convex code in the plane can be realized by interiors of polygons. We give factorial-type bounds on the number of vertices needed to form such realizations. Consequently we show that there is an algorithm to decide whether a convex code admits a closed or open realization in the plane.